Geology Reference
In-Depth Information
Box 8.
D
D
k
k
n
−
1
x
−
x
x
−
x
x
−
x
x
−
x
p
( )
t m x
−
( )
t
(19)
n
−
1
n
−
2
n
−
1
n
n
−
1
n
−
2
n
−
1
n
n
n
−
1
n
−
1
n
−
1
=
0
x
−
x
0
x
−
x
p
( )
t m x t
−
( )
n
n
−
1
n
n
−
1
n
−
1
n
n n
n
Box 9.
D
i
D
i
D
i
k
i
k
i
k
i
−
−
2
1
x
i
−
x
i
x
i
−
x
i
0
x
i
−
x
i
x
i
−
x
i
0
p
i
( )
t
−
m
i
x
i
( )
t
−
2
−
3
−
2
−
1
−
2
−
3
−
2
−
1
−
2
− −
2
2
=
0
x
i
−
−
x
i
x
i
−
x
i
0
x
i
−
−
x
i
x
i
−
−
x
i
p
i
( )
t
−
m
i
x
i
( )
t
1
−
2
−
1
1
−
2
1
−
−
2
1
−
1
− −
1
1
(20)
The number of unknown system parameters
here is
s
= 6. If damage is qualitatively isolated
in one of the columns of the top two floors, the
equations governing the motion of the substructure
containing the top two floors are given in Box 8.
In this case,
s
reduces to 4. If the damaged
columns are located between the third floor and
the (
n
-3)th floor, the equations of motion that
describe the damaged substructure are given in
Box 9.
Herein, s = 6 and
3
is the discrete number of time instants and
∆
t
is
the time step, Eq. (21) can be rewritten as:
[
]
[
]
=
[
]
A
( )
t
R
p
( )
t
(22)
(
2
)
1
(
2 1
)
l
× ×
s
s
×
l
× ×
Alternatively, Eq. (22) can be expressed as:
s
∑
A R
=
p j
;
=
1 2
, ,...,s
×
l
(23)
ji
i
j
i
=
1
i n
. Equations
(18), (19) or (20) can be written in matrix form
as:
≤ ≤ −
2
Assuming
R
i
is the predictor of the ith param-
eter
R
i
, the total error in this estimate is given
by:
[
]
[
]
=
[
]
A
( )
t
R
p
( )
t
(21)
2
s
s
1
2 1
×
×
×
2
is l
×
s
ˆ
]
∑
[
∑
ε
=
p
−
A R
(24)
where,
A
(
t
) is a matrix of system responses,
R
is
a vector of the unknown system stiffness and
damping parameters,
p
(
t
) is a vector of the exter-
nal excitation and the inertia forces and s = 4 or
6. Considering that the structural response is
measured for a time duration
T
j
ji
i
j
=
1
i
=
1
To minimize the total error,
ε
is differentiated
with respect to
R
j
and is set to zero which leads
to:
= ∆
where l
l
t
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